JOURNAL
OF COMBINATORIAL
Series A 59, 218-239 (1992)
THEORY,
Some Remarkable
Combinatorial
Matrices
PHIL HANLON *
Department
of Mathematics,
University
of Michigan,
Ann Arbor, Michigan
48109-1003
Communicated
by the Managing
Editors
Received January 9, 1990
In this paper we describe, in combinatorial terms, some matrices which arise as
Laplacians connected to the three-dimensional Heisenberg Lie algebra. We pose the
problem of finding the eigenvalues and eigenvectors of these matrices. We state a
number of conjectures including the conjecture that all eigenvalues of these matrices
are non-negative integers. We determine the eigenvalues and eigenvectors explicitly
for an important subclass of these matrices. 0 1992 Academic Press. Inc.
1. INTRODUCTION
For r a non-negative
(with 0 = 4).
integer let r0 = (0, 1, .... r} and let r = (1, 2, .... r)
DEFINITION 1.1.1. Let a, b, and k be non-negative
integers with
a 6 k + 1 and b <k + 1. Let Q,(a, b) be the set of pairs (U, V) such that I/
is an A-subset of k, and V is a B-subset of k,. Define the weight of a pair
(U, V) to be
W(U,
V)= c u+ 1 u.
utu
VCV
Let Q,(a, b, w) be the set of pairs (U, V) such that W( U, V) = w.
EXAMPLE 1.1.2. Let a = b = 2 and w = 4. The sets SZ,(a, 6, w) are given
below for every value of k:
(A)
(B)
(C)
k 2 3, &(a, 6, w)= ((03, Ol), (12,01), (O&02), (01, 121,(OLO3)).
k = 2, SZ,(u, b, w) = ((12, Ol), (02,02), (01, 12)).
k=O, 1, O,(u, b, w)=#.
* This work was partially supported by the National. Science Foundation,
Security Agency, and the Cray Research Foundation.
218
0097-3165192 $3.00
Copyright
All rights
0 1992 by Academic Press. Inc.
of reproduction
in any form reserved.
the National
REMARKABLECOMBINATORIALMATRICES
219
For each four-tuple (k, a, b, w) we will define a matrix T,(a, b, w) with
rows and columns indexed by Q,(a, b, w). To define Tk(u, b, W) we need a
notion of when two elements of Qk(u, 6, w) are neighbors.
DEFINITION 1.3. Let (U, V) and (X, Y) be elements of S~,(U, b, w), and
let (u, u, z) be a triple with u E U, v E I’, and z E Z. We say that (U, V) and
(X, Y) are (u, o, z)-neighbors if
(1)
~=(U\bwJ
b+z>
(2)
y=
C-4
(3)
u+v<k.
(~\(+J
In other words, (17, V) and (X, Y) are (u, u, z) neighbors if X is obtained
from U by replacing u with u + z, Y is obtained from V by replacing B with
v-z, and the sum (U + v) = (U + z) + (v-z) of the effected elements does
not exceed k.
EXAMPLE 1.4. (A) (134,05) and (123,25) are (4,0, -2) neighbors for
k B 4.
(B) (123,3) and (024,3) are not neighbors since the size of U n X is
a - 2.
(C) Let k = 3. Then (123,12) is a (1, 1, 0), (1,2,0), and (2,1,0)
neighbor to itself.
The relation of being (u, V, z) neighbors is not a reflexive relation. However,
if (U, V) and (X, Y) are (u, v, z) neighbors then (X, Y) and (U, V) are
(u + z, v - z, -z) neighbors.
The last thing
relation.
we need is a sign associated to each neighborhood
DEFINITION 1.5. (A) Let u= (ul, .... u,) and x= (xi, .... x,) be sequences. Define E(U, x) as follows:
1. E(U, x) = 0 if either u or x has repeated elements.
2. If the elements of u and x are distinct, let 0 and t be the elements
of S, such that
Define E(U, v) to be sgn(a) sgn(z).
(B) Next let U and X be A-subsets such that 1U n XJ > a - 1. In this case
we can write U = {u,, .... u,} and X= { ul, .... ui + z, .... u,} for some z E Z.
Define E( U, X) = E(U, x), where u = (ul, .... u,) and x = (ul, .... ui + z, .... u,).
220
PHIL
HANLON
(C) Lastly suppose that (U, V) and (X, Y) are (u, v, z)-related. Define
&((U, J% (X 0) to be
4( u, v, (X Y)) = 4 u, J3 4 K Y).
It is important to check that the definition of E( U, X) is independent of the
order {ul, .... u,} in which U is written. Also note that E( U, U) = 1 for all
U (here we put z = 0).
EXAMPLE
1.6. (a) Let (U, V) = (156,246) and (X, Y) = (456, 126).
Then (U, V) and (X, Y) are (1,4, 3) neighbors and this relation has sign
- 1. The values of a and /I are 0 and 1, respectively.
(b) Let (U, V) = (12, 12) and (X, Y) = (01,23). Then (U, V) and
(X, Y) are (2, 1, -2) neighbors and this relation has sign + 1. In this case
a=fl=l.
(c) Suppose (U, V) and (X, Y) are (a, v, z) neighbors and that this
relation has sign E. Then (X, Y) and ( U, I’) are (U + z, v - z, - z) neighbors
and this relation also has sign E. The values of a and j3 are the same for
both relations.
(d) The sign of any neighborhood relation between a pair (U, V) and
itself is + 1.
In view of observations (c) and (d) in Example 1.6 the following definition makes sense.
DEFINITION
1.7. Define the graph G,(a, b, W) to be the undirected
signed graph which has an edge between (U, V) and (X, Y) whenever these
pairs are (a, v, z) related. The sign of this edge is E(( U, V), (X, Y)).
EXAMPLE
1.8. (A) Let a = b = 2, w = 4 and assume k is at least 4. The
set Q,(a, b, w) is given in Example 1.2 and the graph G,(2, 2, 4) is
40
-1
(03,011
\A+)/
+I
;?/
(OLO3)
40
(01, 1:)4
((12,02)\tt
-1
+1
(12,Ol)
04
REMARKABLE
COMBINATORIAL
221
MATRICES
Note that at each point we have consolidated the four loops labeled + 1
into a single loop labeled +4.
(B) Let a= b = 2, w =4 again but this time assume k = 3. Then
certain of the neighborhood relations are not allowed; hence certain of the
edges in the above graph must be removed. The graph G,(2,2,4) appears
below:
30
-1
(03901)
(01,lY;
3
+1
/
+1
+1
(02s2J\
(OLO3)
(12901)
-1
30
DEFINITION
1.9. Define the matrix
matrix of the signed graph Gk(a, b, w).
EXAMPLE 1.10. The matrices
respect to the ordered basis
04
T,(a, b, w) to be the adjacency
T,(2,2,4)
(ka4)
and T,(2,2,4)
{ (03,011, (Et01 ), (02,02), (OL12L (OLO3))
are
01-l
1
T,(2,2,4)=
-1
1 4
11
l-11
1
1
4
0
00-I
1
0
4
(ka4),
0
T,(2,2,4)=
-1
11
O-10
(C)
For k = 2, the basis 52,(2,2,4)
{ (12,01), (02,02),
4
0
0
3
is reduced to
(01,
12)).
with
222
PHILHANLON
The matrix T,(2,2,4)
is
The main problem addressed in this paper is the following:
1.11. Determine the eigenvalues and corresponding
matrices T,(a, b, w).
eigenvectors of the
At first glance this problem seems intractable. However, the following
conjecture, which is based on computational
evidence and some special
cases, makes the problem seem more reasonable.
Conjecture 1.12. The
non-negative integers.
eigenvalues
of the matrices T,(a, b, w) are
EXAMPLE 1.13. In this example we will give the eigenvalues and
associated eigenvectors for the matrices T,(2,2,4) which were given in the
last example.
(A) k > 4. The eigenvalues are 2,2,4,6,6.
Below we use independent
eigenvectors associated to each of these eigenvalues:
2
2
4
6
6
(B) k = 3. This time the eigenvalues are 2,2, 3,4, 6. Below we see
independent eigenvectors associated to each of these eigenvalues:
2
2
3
4
6
Observe the interesting change in eigenspaces when we went from the k > 4
REMARKABLE
COMBINATORIAL
MATRICES
223
case to the k = 3 case. The eigenspaces corresponding to 2 and 4 remain
intact but the eigenspace of T,(2, 2,4) corresponding to 6 splits in two
with the two halves becoming the eigenspaces corresponding to 3 and 6,
respectively.
(C) k = 2. The eigenvalues are 5,2, 2. The corresponding eigenvectors are
2
2
5
(-9(-3(i).
In Section 4 we have tables containing
values of a, b, w, and k.
lists of eigenvalues for various
If we suppress the parameter w then we can give a very elegant expression for the eigenvalues. For each a, b, w, k and each non-negative integer
I let ~~(a, b, w; r) denote the multiplicity
of r as an eigenvalue of
T,(u, b, w). Let
/da, b; r) = c pL,(a, b, w; r).
w
Also for i< k+ 1 define p,Ji, 0; 0) and ~~(0, i; 0) to be 1. Lastly, let
M,(x, y, A) be the following generating function for the numbers /~~(a, b; r):
M,(x,
Based on computational
following simple form.
Conjecture
y, 2) = 1 ~,(a, b; r)x*$‘A’.
a,b,r
evidence we conjecture that M,(x, y, A) has the
1.14. M,(x, y, A) = nf==, (1 +x + y + Aci+ “xy).
We would like to improve Conjecture
for the generating function
Mk(x,
y, z, I) =
1
1.14 by giving a nice expression
~,(a, b, w; r)xCIJlbzWAr;
~1,b, n’, r
from the data available we do not see what such an expression would be.
However, we can do that for the coefficient of ;1’ in M,(x, y, z, A). For each
a, b, w let nk(u, b, w) denote the dimension of the nullspace of T,(u, b, w)
and let NJx, y, z) be the generating function
N,(x,
y, z) = 1
o.6.w
582a/S9/2-5
n&z, b, w)xaybzH’.
224
PHIL
HANLON
Note that N,Jx, y, z) is the constant term (with respect to A) of
Mk(x, y, z; 1). Also, if Conjecture 1.14 holds it follows that
N/&Y, y, 1) = (1 + x + Y)& + l.
(1.15)
This expression (1.15) suggests that we try to write N,(x, y, z) as a sum,
over (k + 1)-sequences 0 of O’s, l’s, and 2’s, of
x/l(u) Yi2(o)zwd 9
where j,(o)
function.
is the number of i’s in (T and W(c) is an appropriate
weight
DEFINITION 1.16. Let cr = (a,, cI, .... ok) be a sequence of O’s, l’s, and
2’s. Define W(a) as
W(u)=
i
E;(O)
i=O
where
0
~~(a)=
i
i i+I{j<i:ai=2)1
if
if
if
ai=O
oi=2
bi= 1.
Another way to express W(a) is that W(a) is the sum of the positions in
G which have a nonzero entry plus the number of times a 2 precedes a 1
in U.
For example, if u = 10 2 2 10 2 then
&o(U) = 0
El(U) = 0
&z(U) = 2
E3= 3
c4=6
Es(U) = 0
~4~) = 6,
so W(a) = 17.
Conjecture
1.16. N,(x, y, z) = C, xjl@)yk(“)z W(u’.
REMARKABLE
COMBINATORIAL
225
MATRICES
As an example of this, let k = 1. There are nine sequences d given in the
table below, along with the corresponding monomial x~‘(~)~~*(~)z~(“).
00
10
1
x
20
01
11
21
02
12
22
Y
XZ
x2z
xy2
YZ
xyz
JJ2Z
One can check using the data in Section 4 that this is the correct polynomial N,(x, y, z). At present the author does not have a conjecture similar
to Conjecture 1.16 for the whole polynomial Mk(x, y, z; 1).
2. STABILITY
For certain sets (a, b, w, k) we do know the eigenvalues and eigenvectors
of the T,(a, b, w). Before stating the condition which defines these sets it is
worth noting that Q,(a, b, w) is empty unless w is at least (f) + (i). Also the
matrix T,(a, b, w) is similar to the matrix T,(b, u, w), so when we consider
these matrices we may assume that a < 6.
DEFINITION 2.1. Let (a, b, w, k) be a four-tuple
w > (;) + (g). We say this four-tuple is stable if
(1)
(2)
with
a ,< b and
w<(;)+@+a.
k>(a-l)+(b-l)+(w-(z)-(i)).
PROPOSITION 2.2. Let (a, b, w, k) be a stable four-tuple with a < b and
n = w - (z) - (i). Then .CJk(a,b, w) is in l-l correspondencewith the set of
pairs (,I, p), where A and p are partitions of p and n - p for somep d n.
Proof This
exact bijection
follows:
Let 1= (A,,
p and (n-p),
corresponding
u= {a-
follows easily from condition (1) of Definition 2.1. The
between Q,(a, b, w) and the set of pairs (2, ,u) is given as
&, .... 4) and P = hr
so that l<p<n<a
set (U, V) to be
1 +&,a-2+1,,
V={b-l+p,,b-2+pZ
pz, .... II,) be a pair of partitions of
and m<n-p<n<b.
Define the
...) a-f+l,,a-l,..., b-m+p,,,,b-m-l,...,
1, ...) 0)
O}.
1
226
PHIL
HANLON
The main result of this section determines the eigenvalues and eigenvectors explicitly. To state this result we need some notation from the
representation theory of the symmetric and hyperoctahedral groups. Let
(a, b, W, k) be a stable four-tuple and let {x,, .... x,, y,, .... yb} be a set of
commuting indeterminates. Let {z I , .... z, + h} be a second set of commuting
indeterminates. Symmetric functions notation follows [M].
For each YE N let p,?(x, y) and p; (x, y) denote
Pr(x+Y)=
E x:+
i=
PAX-Y)=
i
xi-
i=l
For A=i,,I,
... A,, a partition,
i
j=
I
i
Y,'
1
y;.
i=l
define pi(x + y) by
PAX + Y) =
n Pi&X + Y)
.Y=l
PAX-Y)=
n
P&(x-Y).
S=l
Lastly, if p is a partition of n define SJX + y) as follows: first write the
Schur function s,(zr, .... z,,~) in the form
Then
s,(x+Y)=p:P,(x+Y)
1
and
The signed Schur functions sP(x + y) and the signed power sums
pi(x + y) originally came up in the representation theory of the hyperoctahedral group. The two sets
(sJ,(x + Y) sp(x - Y): I4 + IPI = n> = %
and
{Pi(x+Y)PJx-Y):
I4 + IPI =n> =a*
REMARKABLE
COMBINATORIAL
MATRICES
227
each form a Q-basis for the homogeneous polynomials of degree n that are
symmetric in the x’s and symmetric in the y’s. The transition matrix
between these two bases is the character table of the hyperoctahedral
group H,.
There is a more natural basis for the same set of homogeneous polynomials, that being
{Tdx) s,(Y): 14 + IPI = 4 = g3*
We will make use of the transition
basis a, above.
matrix between this basis 9J3 and the
DEFINITION 2.3. Detine II{;:“,\ for all (a, /I), (A, p) with
111+ 1~1=n by the equation
&(X + Y1$6
- Y) = c $:$(x)
CAP)
Ial + I/?! =
S,(Y ),
EXAMPLE 2.4. (A) Suppose fl= 4 so a is a partition
of n. Then
X,(X + y) sa(x - y) = s,(x + y) = s,(xr, .... x,, y,, .... yb). By standard symmetric function arguments we have
&(X9Y) = c b?,,%(X) S,(Y),
J.>P
where gnPa is the number
content p. So we have
of Littlewood-Richardson
fillings of a/d with
(B) Suppose
a =d
so p is a partition
of n. Then
s,(x + y) ss(x - y) = sB(x - y). It is not hard to apply the arguments from
(A) above to show that
(C)
Let
n = 2. Cases (A) and (B) above determine
I$;~{ for
values
(a, 8) = (2, b), ( 12, $), (4, 12), and (4,2). To compute the remaining
I$$‘, note that:
228
PHIL
HANLON
So the values of z$$i are u$;:l= u;:$‘;, = 1, ui:,:l=O,
and u$&=
(Ll) - - 1. A complete listing of the v[T$
,
for
n
=
2’
appears
in the table
uM2J below:
(A pc)\(a,8)
(234)
(I224)
(234)
1
0
1
0
I
(17
0
1
1
1
0
(41)
1
1
0
(4, 17
0
1
-1
0
1
2)
1
0
-1
1
0
($4
(1, 1)
(d?19
-1
(432)
-1
The reader should compare the columns of the above table to the eigenvectors for T,(2, 2, 4) given in Example 1.13 (A).
THEOREM
2.5. Let
(a, b, w, k)
be a stable four-tuple
and let
n = w - (z) - (i). Let 9n be the set of pairs of partitions (I, p) such that
IAl+ IpI =n. Assume 9” is ordered in some way and that T,(a, b, w) is
constructed according to that ordering (here we have identified 9” with
&?,(a, b, w)). Then
(a) for each (CC,/I)E~~ the vector (u$~{)(~,~~~~~ is an eigenuector of
T,(a, b, w) with corresponding eigenvafue 1~1- IpI + ab.
(b) The eigenvectors constructed in part (a) are linearly independent;
hence they give a complete set of eigenvectors.
Proof. Condition (2) in the definition of stability implies
is a pair in SZ,(a, b, w) and u E U, u E V then u + u 6 k. This
is large enough so that we never need to consider condition
tion 1.3 when we determine the entries of T,(a, b, w). In
diagonal entries of T,(a, b, w) are all equal to ab. Let M be
M=
that if (U, V)
means that k
(3) of Deliniparticular the
the matrix
T,(a, b, w) - (ab)l.
We will determine the eigenvalues and eigenvectors of M acting on the
basis Mx) Q(Y)).
Let n(x) be the ring of symmetric functions in xi, .... x,. For each r,
multiplication
by the power sum symmetric function p,(x) constitutes a
linear transformation of n(x). To avoid confusion we denote this linear
transformation by P,(x). Our immediate goal is to express M in terms of
these transformations P,(x).
REMARKABLE
COMBINATORIAL
229
MATRICES
Recall the following notation from the theory of symmetric functions. If
Y = (Y, 3 ..*, y,) is any non-negative integral sequence then sT(x) denotes
det(x?+“-i)
qx) = det(x;-‘)
= E(Y + PI Jdy + p, - p(x),
where p = (a - 1, a - 2, .... 0) and (T is the permutation
entries of y + p in decreasing order.
in S, which puts the
LEMMA
2.6 [M, p. 321. Let II be a partition of length less than or equal
to a. Write I = (A,, .... A,), where A, > A2 > .-. > A,2 0. Let r be a positive
integer. Then
P,(xMx))=
i
sk+re,(x)>
i=l
where e,, .... e, are the unit coordinate vectors in h”.
Let ( , ) be the usual inner product on the ring A(x). So we have that
the Schur functions are an orthonormal
basis of A(x) with respect to
( , ). The power sum symmetric functions satisfy
(PAX)? P,(X)> = Z,~A,?
where zA is the order of the centralizer in SIA, of a permutation with cycle
type I. Let P,*(x) denote the adjoint of P,(x) with respect to ( , ). So
P,*(x) is defined by
<Rvx)
From Lemma
have
4x1, u(x)> = (4x1, P,(x) v(x)>.
2.6 and the fact that (sI(x)}
is an orthonormal
f?(x)s,(x)=c s,-rqb).
basis we
(2.7)
On the right-hand side of (2.7) the sum is over those i such that pj - rei > 0.
We will need to rewrite Lemma 2.6 and (2.7) in slightly different forms.
Let A = (A.,, .... A,) be a partition of f where f < a. Let U be the a-set
(n, + a - 1,122+ a- 2, ...) A,> = (U1) ...) u,> and let r be a positive integer.
Fix i, let A?+) denote the set {u,, .... ui+r, .... u,> and let XC-’ denote the
set (ur, .... ui-r, .... u,>. Let p(+) and p(-) denote the partitions off + r
and f - r corresponding to XC+ ) and A’(-‘, respectively. Then
the coefficient of sP,+,(x) in P,(x) sA(x) equals E(U, A?+‘)
the coefficient of sp,-,(x) in P,*(x) s).(x) equals E(U, XC-‘).
(2.8 1
230
PHIL
HANLON
From (2.8) it follows that M represents the linear transformation
M=
f
P,(x) WY) + P,*(x) PAY)
r=l
on V, = (n(x) @/i(y))‘“‘, where the subscript (n) denotes the subspace of
A(x
of polynomials which are homogeneous of degree n. Interestingly enough, the same linear transformation M comes up in the work
of Stanley on r-differential posets (see [St]).
There is a second basis for V,, namely the set {s,(x + y) sB(x - y):
Ial + Ifi1 =n}. Let V,“”
( ) denote the span of all s,(x +y) s&x-y),
where
Ial = r and 181= s. We will show:
1. M preserves the space VP”‘.
(2.9)
2. M restricted to VF”’ is just the scalar matrix r-s.
To check (2.9) we can apply M to any basis of VF”’ that we like. Let
4?~‘) denote the basis {p,(x +y) p&x-y):
la/ = Y, IfiI =s}. Note that
P,(x) P,*(v) is a derivation so M is a derivation as well. It follows that
It remains to show that Mp,(x+y)=
IaJ pJx+y)
- 181ps(x - y). Again using that M is a derivation,
MPa(x+Y)=
i
and MpB(x-y)=
we have
(2.10)
(~P&+YH
i=l
We must determine Mp,(x
(WY)
+ y) = M. (p,,(x)
+p%,(y)).
Note that
PAY), P,(Y)>
= (PAY),
P,(Y) P,(Y)>
if
if
0
= t (PAY)? PAY) P,(Y)>
rfs
r=s
unless r =s, ~L=c#
if r=s, p=d
=
r
0
if r=s, p=qS
otherwise.
so kf* (PAX) +Ps(Y)) = P,(Y) P,*(x) PAX) + P,(x) f?(Y)
p,(x)). By (2.11) we have M.p,(x + y) = loll p,(x + y).
(2.11)
PAY) = dP,(Y) +
REMARKABLE
COMBINATORIAL
231
MATRICES
Similarly, M. (P,(X) -P,(Y)) = v,(y) -v,(x) = (-s)(p,(x) -p,(Y)). So
MT&-Y)=
-IPI
&3(X-Y).
This proves (2.9) which completes the proof of Theorem 2.5. 1
Theorem 2.5 tells us what happens when a and b are large with respect
to w, and k is large with respect to a and b. Example 1.13 shows how the
eigenvalues and eigenvectors can change when this ideal situation
destabilizes by k becoming small. We now do one example where
destabilization occurs by a and b becoming small. Begin with the stable
parameters a = b = 2, w = 4, k > 4. As a and b become small we will change
w to preserve n=w-(z)-(i).
EXAMPLE 2.12. (A) Let a = 1, b = 2, w = 3, and k= 3. We have
a,( 1, 2, 3) = ((2,01), (1,02), (0,03), (0, 12)). The matrix T3( 1,2, 3) with
respect to that ordered basis is
2
1 1
121
112
i -110
1
0’
2!
The eigenvalues and corresponding
eigenvalue
A
-1
eigenvectors for this matrix are
0
1
3
4
(B) Keeping u = 1, b = 2, let k = 2. The ordered basis Q,(l, 2, 3) is
((2, 01), (1,02), (0, 12)). The matrix T,(l, 2, 3) is
7’,(1,2,3)=
The eigenvalues and eigenvectors are
232
PHILHANLON
(C)
Lastly,
((2, Oh (1, 11, (0,2)}
let a=6=1,
and
The eigenvalue and corresponding
w=2,
and k=2.
Then
.R,(l, 1,2)=
eigenvectors are
By considering extreme values of a, b, and k it is possible to make a
number of specialized conjectures about how this destabilization occurs.
But the general procedure remains quite mysterious.
3. LAPLACIANS OF NILPOTENT UPPER SUMMANDS
The matrices &(a, 6, w) arise in an algebraic setting. In this section we
will briefly describe their algebraic interpretation
and we will state a
conjecture about Laplacians
of nilpotent
upper summands which
generalizes Conjecture 1.14.
Let 9 be a semisimple complex Lie algebra and let H be a Cartan
subalgebra. Let R be the root system of Y and let
be the root space decomposition of 9.
For S a set of simple roots, let R, be the root system of R generated by
S. Let ?& be the semisimple subalgebra of 9 given by
DEFINITION 3.1. For S a set of simple roots let N’s’ be the nilpotent
algebra given by
NCS’=
Q
cc&.
reR+\RS+
We call NC’) a nilpotent upper summand of ‘9.
Lie
233
REMARKABLECOMBINATORIALMATRICES
EXAMPLE 3.2. Let A?’ be the three-dimensional
Heisenberg Lie algebra.
So A? has C-basis (e, f, x} with non-zero brackets
[e,f]=
-[f,e]=x.
Then X is the nilpotent upper summand of 9 = S/~(C) obtained by taking
S = 4. The isomorphism is given by sending zIz to e, z13 to f, and .z13to x
(where zij is the matrix in SIP
which has a 1 in the i, j entry and O’s
elsewhere).
DEFINITION
non-negative
3.3. Let N be a complex Lie algebra and let k be a
integer. Define N, to be the Lie algebra
Nk = N@ {C[t]/(t”+‘)}.
The bracket in Nk is given by
[xOa(t), yOb(f)l= Lx, rl@Mf)
b(f))*
Let M be any complex Lie algebra. For each r, define 8,: A’M + A’- ‘A4
by
c?‘(m, A ...
A nz,)
= C (-l)i+i-l
[mi,mj]
Am, A . . . *$A
... *GA
... Am,.
i-zj
It is straightforward
to check that a,-, od,=O.
H,(M)
Define H,(M)
by
= ker a,/im d,, 1.
H,(M) is called the homology of M.
Suppose, in addition,
that M is a graded Lie algebra, i.e.,
M= @ TCOM,, where CM,, M,] c MU+ “. In this case we introduce a
second grading on AM (which we call weight) as follows: if mi E M, then
the weight of m, A .a. A m, is (si + . .. + s,). Let A’,“M denote the span
ofallm,
r\ .-. A m, which have weight w.
It is easy to see that 8, preserves weight, i.e.,
a,: A’*“M+
A’-‘,“M.
This implies that the notion of weight passes to the homology of A4 and we
obtain a subsequent bigrading of H(M).
Let ( , ) be a Hermitian
form on A4 with the property that
234
PHIL
HANLON
(M,,M,)=O
if u#v. Then ( , ) gives rise to a Hermitian
(also denoted ( , ) ) by
(ml A ... Am,,p,
A ... Ap,)=
0
det((m,,
P,>)
form on AM
if
if
r#s
r = s.
It is easy to see that
(A”,“lM,
,,.“2M)
=
unless
0
ri = y2 and wi = w2.
3.4. Define 8: to be the adjoint of 8, with respect to ( , )
and define L to be
DEFINITION
L=
aal+ ata.
We call L the Laplacian of M.
Note that 8’ and L both depend on ( , ). In practice the form ( , )
will be understood and so we will not refer to ( , > when we talk about
a’ and L. It is easy to see that
so L: Ai’,“M+
A’,“M.
It is straightforward
to check that a:+ i a:=O. Define H*(M)
by
= ker aijirn ai_ 1.
H*(M)
H*(M)
is called the cohomology of M. As the following result indicates, the
kernel of the Laplacian is of particular interest.
THEOREM
3.5 (see [K] ). Any basis for the kernel of L simultaneously
gives a complete set of homology representatives and cohomology representatives.
For the rest of this section we will look at a particular graded Lie
algebra M. Let 2 = (e, f, x) denote the three-dimensional
HeisenbergLie algebra and let M = Y$. Grade M by
*
u
=
X@tU
0
if u<k
otherwise.
For notational convenience we let e,, f,,, and x, denote e@ P, f@ t”,
and x 0 t”, respectively.
REMARKABLE
Let ( , ) be the Hermitian
respect to the basis g’,
COMBINATORIAL
235
MATRICES
form whose Gram matrix is the identity with
B= {e u : u = 0, 1, “.) k} u {fu : 24= 0, 1, ...) k} u {x, : u = 0, 1, .... k}.
We henceforth assume that d’ and L are defined with respect to the form
( , >.
Let E, F, and X be the subspaces of A4 spanned by the cur f,, and x,,
respectively. We have A4 = E 0 F@ X so
AM=
(AE) @ (/IF) @ (AX).
(3.6)
We need an observation concerning the way d behaves with respect to the
decomposition (3.6). The only non-zero brackets in M are of the form
Ce,,LJ
=-*,+,.
so,
a:(A”E)@(AbF)@(A’X)+(AO-‘E)@(A”-’F)@(A’+’X).
(3.7)
Using (3.7) it is easy to see that
~':(/~~E)O(/~~F)O(/IC~)~(~~+~E)O(/~~+IF)O(/~~-JX).
Combining
(3.8)
(3.7) and (3.8) we have that
L: (A”E)@(AbF)@(AcX)+(A”E)@(AbF)@(/lcX).
Let V,(a, b, c; w) denote the subspace of AM given by
V,(a, b, c; w) = (A**“(M))
n { (A”E) @ (AbF)@ (AcX)}.
(3.9)
From (3.9) and previous observation that L preserves weight, we have
L: V,(a, 6, c; w) + Vk(a, 6, c; w).
(3.10)
We have the following theorem which concerns the map (3.10) just in the
case c = 0.
THEOREM
3.11. (A) We can choose a basis for V,(a, 6, 0; w) which is in
natural bijective correspondence with Qk(a, b, w).
(B) The matrix for L with respect to that basis is L,(a, b, w).
The proof of Theorem 3.11 is not difficult and we leave it to the reader.
One could ask if the restriction of L to V,(a, b, c; w) has a combinatorial
582al59/2-6
236
PHILHANLON
description for any c. The answer is yes but the description is significantly
more complicated when c is nonzero.
As mentioned earlier the three-dimensional
Heisenberg-Lie algebra is
one example of a nilpotent upper summand. At least one conjecture from
Section 1 seems to have an analogue in the more general situation. Let
N= N@) be a nilpotent upper summand of a semisimple Lie algebra $9.
Kostant [K] showed that there exists a positive definite Hermitian form
{ , > on N with the property that the Laplacian L of N with respect to
{ , } has non-negative integer eigenvalues. For each pair of non-negative
integers r, n let m(N; r, n) denote the multiplicity
of n as an eigenvalue of
the restriction of L to .4’N. Define the generating function E(N, z, 2) to be
the generating function for the numbers m(N; r, n), i.e.,
E(N; z, A) = C m(N,; r, n)z’A”.
i-.n
Define a positive definite Hermitian
(XQ t’, yo t’) =
form ( , ) on Nk by
{x3Y>
i0
if
if
i=j
i# j.
Let Lk denote the Laplacian of Nk with respect to ( , ) and define
m(N,; r, n) as above to be the multiplicity
of n as an eigenvalue of the
restriction of Lk to AINk. Also define E(Nk; z, A) by
E(N,; z, A) = 1 m(N; r, n).z’jl”.
r,”
We end with the following conjecture which generalizes Conjecture 1.14
from the three-dimensional
Heisenberg to arbitrary nilpotent
upper
summands.
Conjecture 3.12. With notation
as above we have
kfl
E(N,;
z,
1)
=
n
E(N,;
Z, Ai).
i=l
Conjecture 3.12 is one of a series of conjectures made by the author
concerning the homology of N, for various Lie algebras N. The reader can
find a thorough discussion of these conjectures in [H].
4. COMPUTATIONAL
RESULTS
Below we see lists of the eigenvalues of the matrices Tk(a, b, w) for some
Of k. since Tk(a, b, W) = Tk(b, a, W) we Only list those triples
Small VdUfX
REMARKABLE
COMBINATORIAL
MATRICES
237
with a < b. Also if a = 0 then the matrix TJO, b, W) is the p xp niatrix of
O’s, where p is the numbers of partitions of w with length b. So we also omit
the case a = 0 on our list.
k=l
k=2
a
b
M’
Eigenvalues of T,(a, b, w)
1
2
0
1
1
0, 2
2
1
2
1
0
2
2
2
2
2
3
a
b
W’
Eigenvalues of T,(n, b, w)
0
1
1
0, 2
2
3
4
0, 0, 3
030
0
2
1, 3
0,1,3
a 2
1
1
1
3
2
3
4
5
3
4
5
2
3
4
5
6
4
5
6
6
a
w
2
k=3
1
1
0
1
2
3
4
5
6
1
2
1
2
3
4
1
0
3
2
1
4
3, 3
2,2, 5
1, 3
1
5
4
3
6
Eigenvalues of T,(a, b, w)
1
092
(40, 3
0, 0, 0,4
0, 0, 0
0, cl
0
2
1. 3
0, 1, 3,4
0, 0, 1, 2, 4
238
k = ~-CON.
PHIL
a
w
b
Eigenvalues
5
1
3
6
I
8
3
4
5
6
1
1
4
2
2
2
3
2
4
3
3
3
4
4
4
HANLON
8
9
6
I
8
9
2
3
4
5
6
I
8
9
10
4
5
6
I
8
9
10
11
7
8
9
10
11
6
I
8
9
10
11
12
9
10
11
12
12
of T,(a, b, w)
0.0.0,2.4
o,o, 0, 3
0, 0
0
3
2.4
1,2,4
0, 1, 3,4
Rl.3
0,2
0
4
3
2
4
3, 5
2, 2, 3,4, 6
L2, 2,4,4, 5
0, 1, 1, 3, 3, 3,4, 7
0, 1, 1, 3, 3,4
0, 0, 1, 294
0, 2
0
6
5, 5
4,4,4,1
3, 3, 3, 5, 7
2. 2, 334, 6
1. 2.4, 5
1, 3
1
7
6
5, 5
4
3
8
7, 7
66, 6
5, 5, 5,9
4,4, 7
3, 5
3
9
8
7
6
10
REMARKABLECOMBINATORIALMATRICES
239
ACKNOWLEDGMENTS
I thank Professor Ian Macdonald for his help with this work and in particular for help with
the proof of Theorem 2.5.
REFERENCES
P. HANLON,
Some conjectures and results concerning the homology of nilpotent Lie
algebras, Adu. Math. 84, No. 1 (1990), 91-134.
[K] B. KOSTANT, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of
Math. 74 (1961), 329-387.
[M]
I. G. MACDONALD,
“Symmetric Functions and Hall Polynomials,” Oxford Univ. Press,
London, 1979.
[St] R. P. STANLEY, Differential posets, J. Amer. Math. Sot. 1, No. 4 (1988), 919-961.
[H]